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21: 4.43 Cubic Equations

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22: 16.23 Mathematical Applications

… ►In Janson et al. (1993) limiting distributions are discussed for the sparse connected components of these graphs, and the asymptotics of three ${}_2{}^{}F_2^{}$ functions are applied to compute the expected value of the excess. …

23: 20.14 Methods of Computation

… ►For instance, the first three terms of (20.2.1) give the value of ${\theta }_1\left(2-\mathrm{i}\right|\mathrm{i})$ ($={\theta }_1(2-\mathrm{i},{\mathrm{e}}^{-\pi })$) to 12 decimal places. …

24: 22.1 Special Notation

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, $\mathrm{dn}(z,k)$; the nine subsidiary Jacobian elliptic functions $\mathrm{cd}(z,k)$, $\mathrm{sd}(z,k)$, $\mathrm{nd}(z,k)$, $\mathrm{dc}(z,k)$, $\mathrm{nc}(z,k)$, $\mathrm{sc}(z,k)$, $\mathrm{ns}(z,k)$, $\mathrm{ds}(z,k)$, $\mathrm{cs}(z,k)$; the amplitude function $\mathrm{am}(x,k)$; Jacobi’s epsilon and zeta functions $ℰ(x,k)$ and $\mathrm{Z}\left(x|k\right)$. …

25: 24.19 Methods of Computation

… ►We list here three methods, arranged in increasing order of efficiency. …

26: 26.2 Basic Definitions

… ►The example $\{1,1,1,2,4,4\}$ has six parts, three of which equal 1. …

27: 34.6 Definition: $\mathit{9}j$ Symbol

… ►The $\mathit{9}j$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

28: 19.1 Special Notation

… ►The first three functions are incomplete integrals of the first, second, and third kinds, and the $\mathrm{cel}$ function includes complete integrals of all three kinds.

29: 19.14 Reduction of General Elliptic Integrals

… ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. …

30: 31.7 Relations to Other Functions

… ►Below are three such reductions with three and two parameters. …